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Talk:Ultra-hyper-cosmo-galaxi-cosmo-hyper-ultrol
\(\omega^6+\omega^5+\omega^4+\omega^3+\omega^4+\omega^5+\omega^6=\omega^6+\omega^6\) LittlePeng9 (talk) 18:19, October 10, 2014 (UTC) :It would probably grow at the same rate, but it's not actually equal. -SJ224 18:25, October 10, 2014 (UTC) ::The equality I gave you is exact and holds in realm of ordinal numbers. LittlePeng9 (talk) 18:31, October 10, 2014 (UTC) :::The way I understand the fast growing hierarchy, those two sums would represent the same growth rate, but I'm not sure they're always equal. -SJ224 18:35, October 10, 2014 (UTC) ::::I'm not talking about fast growing hierarchy now. I'm talking about ordinals. The two sides of equality are exactly equal, neither is greater at all. LittlePeng9 (talk) 18:42, October 10, 2014 (UTC) :::::Ordinals in FGH is not just ordinals. Also, the notation is easier to understand and more exact than w^6+w^6. As for ordinals, I think the two sides are equal.--Nayuta Ito (talk) 22:05, October 10, 2014 (UTC) ::::::Ordinals in FGH are ordinals. If they aren't ordinals, then you're not dealing with FGH, which is fine but you need to define this alternate system it's vel 23:46, October 10, 2014 (UTC) :::::::What do you mean by "not just ordinals"? Is there some kind of superordinals which are used here that I don't know of? I'd love to see how these are defined. LittlePeng9 (talk) 05:10, October 11, 2014 (UTC) :::::::me too. Yet again I will clarify that it is totally cool and fine to define a variant of FGH that allows these alternate constructions. But you can't say that your result is FGH, and you certainly can't get away with leaving it undefined. it's vel 06:46, October 11, 2014 (UTC) In fgh as I've understood it (ever since I understood it past omega at all), ordinal sums are reduced one term at a time from right to left. Therefore, beyond some minimum n (almost certainly less than 100), f_{\omega^6*2}(n). -SJ224 10:01, October 11, 2014 (UTC) :Sbiis Saibian has given a definition of FGH that includes "rogue-types". Using that definition, SJ224's inequality holds. -- ☁ I want more ⛅ 10:07, October 11, 2014 (UTC) :SJ224: this isn't really how ordinal sums work in FGH, because the have to be in so called Cantor's normal form, which requires the exponents of \(\omega\) powers to be written in non-increasing order, e.g. \(\omega^3+\omega^2\) is allowed, but \(\omega^3+\omega^2+\omega^3\) isn't. :Cloudy: in that case, he should mention wherever that he doesn't use standard definition of ordinal hierarchy. LittlePeng9 (talk) 11:21, October 11, 2014 (UTC) :SJ: "ordinal sums are reduced one term at a time from right to left" Nope. As Peng pointed out, we're using Cantor normal form, not the "sum of terms." If you look up the definition of FGH, you'll see that the ordinal has to be expressed as a finite decreasing series of powers of \(\omega\). :One nitpick in Peng's comment: the use of Cantor normal form is just according to one definition of a fundamental sequence (specifically the Wainer hierarchy, which is defined for ordinals \(\leq \varepsilon_0\)). There are many alternate definitions, but no matter what, \(f_{\omega^6+\omega^5+\omega^4+\omega^3+\omega^4+\omega^5+\omega^6}=f_{\omega^6+\omega^6}\). That's a consequence of a very basic law of algebra, although somehow people don't seem to comprehend this... :Cloudy: It's concerning that he calls what he's defined "FGH" in that article. It's not. It's not an ordinal hierarchy, but an extension of one. it's vel 16:46, October 11, 2014 (UTC) This value of this number clearly exceeds fω6*2(100), it actually falls somewhere between fω6*2(102) and fω6*2(103) (which, googologically speaking, isn't actually a vast improvement, but it's still one that's worth noting.) —Preceding unsigned comment added by SuperJedi224 (talk • )